Quillen model structures for relative homological algebra
Quillen model structures for relative homological algebra
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DOI：
10.1017/s0305004102006126
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发表时间：
2000-11
影响因子：
0.8
通讯作者：
J. Christensen;Mark Hovey
中科院分区：
文献类型：
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作者：
J. Christensen;Mark Hovey
模型范畴的一个重要例子是 R 模的无界链复形范畴，它的同伦范畴是环 R 的派生范畴。这个例子表明传统的同调代数包含在 Quillen 的同伦代数中。本文的目的是表明同调代数的更一般形式也适合 Quillen 的框架。具体来说，完备且余完备阿贝尔范畴 [Ascr ] 上的射影类正是在 [Ascr ] 中进行同调代数所需的信息。主要结果是，在弱假设下，[Ascr] 对象的链复合体范畴具有一个模型范畴结构，该结构反映了射影类的同调代数，因为它编码了 Ext 群和更一般的派生函子。例子包括环 R 的“纯派生类别”，以及捕获相对情况的派生类别，包括 Hochschild 同源和上同调的投影类。我们描述了共纤维生成的模型结构的特征，并表明这对于许多有趣的例子来说是失败的。最后，我们解释了可能的非阿贝尔范畴中的单纯对象的范畴如何配备反映给定射影类的模型范畴结构，并给出了包括等变同伦理论和有界于派生范畴之下的示例。
An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category [Ascr ] is exactly the information needed to do homological algebra in [Ascr ]. The main result is that, under weak hypotheses, the category of chain complexes of objects of [Ascr ] has a model category structure that reflects the homological algebra of the projective class in the sense that it encodes the Ext groups and more general derived functors. Examples include the ‘pure derived category’ of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and co-homology. We characterize the model structures that are cofibrantly generated, and show that this fails for many interesting examples. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories.